Piatetski-Shapiro’s Work on Converse Theorems
J. Cogdell · Contemporary mathematics - American Mathematical Society · 2014
Converse theorems were a central feature of Piatetski-Shapiro’s work on automorphic L L -functions, from his first paper on the subject in 1971 to the last applications to functoriality in 2011. Converse theorems give criteria, in terms of L L -functions, for a global representation of G L n GL_n to be automorphic; if one views the representation as parametrizing an Euler product, they give analytic criteria for an Euler product to be modular. The converse theorems for G L n GL_n all involve controlling the properties of these L L -functions when twisted by cusp forms on smaller G L m GL_m . The most basic ones require twisting by (essentially) all cuspidal representations of smaller rank groups, either rank up to n − 1 n-1 or up to n − 2 n-2 . These are primarily spectral in nature and are those that have had the most applications. There are also those that significantly restrict the ramification of the twisting representations. These have a significant algebraic or arithmetic component (generation of congruence subgroups) but as yet have no applications that I am aware of. Then there are also the so-called “local converse theorems”. We will survey what is known, what is expected, and how these have been used.